Unbounded spectral minimal partitions can exist at threshold

What the study found: The authors study k-spectral minimal partitions on unbounded domains, including domains of infinite volume. They find that existence and structure of minimizers depend on whether the energy level is below or at a threshold involving the essential spectrum of a Schrödinger operator (an operator of the form -Δ+V) and lower-order partitions.
Why the authors say this matters: The study suggests that new phenomena appear for spectral minimal partitions when the domain is unbounded, and the authors use examples to show how these behaviors can occur for different domains and potentials.
What the researchers tested: The paper analyzes partitions that minimize a p-norm, with 1 le p le infty, of the lowest spectral value of a Schrödinger operator with Dirichlet boundary conditions on each partition cell. The authors prove a sharp upper bound, develop a concentration-compactness-type argument below the threshold, and examine what happens at the threshold for both p < infty and p = infty.
What worked and what didn't: Below the threshold, optimal partitions exist and each cell has a ground state, meaning the lowest spectral value is a simple isolated eigenvalue. For p < infty at the threshold, minimizing partitions may or may not exist, and even when they do, they may lack ground states. For p = infty, minimal partitions always exist, including at the threshold, but they may or may not admit ground states; below the threshold, a minimizer can always be constructed and it is an equipartition, while at the threshold spectral minimal partitions need not be equipartitions.
What to keep in mind: The abstract does not provide a full list of assumptions beyond the spectral setting and the unbounded-domain framework. It also notes that the authors give examples, but the specific examples are not described in the available summary.

Key points

• The paper studies k-spectral minimal partitions on unbounded domains, including infinite-volume domains.
• A sharp upper bound is proved using a threshold tied to the essential spectrum and to (k-1)-partitions.
• Below the threshold, optimal partitions exist and each cell has a ground state.
• For p < infty at the threshold, minimizing partitions may or may not exist, and they may lack ground states.
• For p = infty, minimal partitions always exist, but they may or may not be equipartitions.